(a) State and discuss the important points which have to be considered in the construction of index numbers. (10 marks) (b) In a…

(a) Points to consider in the construction of index numbers

An index number measures the relative change in a variable (price, quantity or value) between a base period and a current period. The main points are: Purpose: the objective decides the type of index, the items and the weights, for example a consumer price index to measure the cost of living. Selection of items: a representative sample of commodities that are important, of standard quality and available in both periods. Price collection: prices from representative markets and sources, for a defined time, with uniform units and quality. Base period: a normal period, free of unusual events such as wars or famine, and not too distant, so that comparison remains meaningful. The base should be revised when consumption patterns change. Choice of average: the arithmetic mean is simple, while the geometric mean is preferred for averaging relatives because it reduces the effect of extreme values and satisfies the time-reversal test. Weights: items differ in importance, so weights such as base-year quantities (Laspeyres) or current-year quantities (Paasche) are used. Fisher's ideal index is the geometric mean of the two. Choice of formula: it should satisfy the tests of consistency: time reversal, factor reversal and, for chain indices, circular test. Limitations: substitution bias, changing quality of goods and new products.

(b) Interpretation of the trend equation y = 1500 + 300x (origin 2000, x in years)

Step 1: (i) The slope is 300, so average annual increase in units sold is 300 units. Step 2: (ii) Setting 1500 + 300x = 4500 gives x = 10, which is the year 2000 + 10 = 2010. Expected sales then equal the present capacity of 4500 units. Step 3: (iii) For 2020, x = 20, so y = 1500 + 300 × 20 = 7500 units. Step 4: (iv) The slope of the trend line is 300 units per year. Step 5: (v) Excess over the present capacity in 2020 = 7500 − 4500 = 3000 units.

(c) Reliability and validity of a test

Reliability is the consistency of a test, meaning that it gives nearly the same scores when repeated under the same conditions; it shows freedom from random error and is estimated by test-retest, parallel-form or split-half methods. Validity is the degree to which the test measures what it claims to measure; it is judged by content, criterion-related or construct evidence. Difference: reliability concerns consistency of the scores, while validity concerns their meaning. A reliable test can be invalid, but an unreliable test cannot be valid, so reliability is necessary but not sufficient for validity. Calculation: when a test is lengthened, its validity coefficient with the criterion becomes r_new = r_old × √(R_new/R_old). Step 1: r_new = 0·60 × √(0·90/0·80) = 0·60 × 1·061. Step 2: r_new ≈ 0·64. (The lengthening factor is n = 0·9 × 0·2/(0·8 × 0·1) = 2·25, and 0·6 × √2·25/√(1 + 1·25 × 0·8) gives the same 0·64.) The validity coefficient is expected to rise from 0·60 to about 0·64.

(d) Population at t = 30

The rate r(t) is taken as the per-capita growth rate, so dP/dt = r(t)P. Step 1: dP/P = (0·01 + 0·0001t²) dt. Step 2: Integrating, ln P = 0·01t + 0·0001t³/3 + C. Step 3: At t = 0, P = 10⁶, so C = ln 10⁶ and P(t) = 10⁶ exp(0·01t + 0·0001t³/3). Step 4: At t = 30, the exponent is 0·3 + 0·0001 × 27000/3 = 0·3 + 0·9 = 1·2. Step 5: P(30) = 10⁶ e^1·2 ≈ 3,320,117. The population at t = 30 is about 3·32 million.

(e) Choice between morbidity incidence and prevalence rates for malaria medicine

Prevalence rate (MPR) counts all existing cases, old and new, at a point of time or over a period, per population. Incidence rate (MIR) counts only new cases arising in a period. Recommendation: MPR should be used to decide the quantity of medicine to send. A drug stock must be enough for all persons who are ill and need treatment at that time, and MPR gives that burden, which also reflects how long patients remain ill (prevalence is about incidence times average duration). Antimalarials, rapid diagnostic kits and supplies are then sized to the number of cases to be treated, with a margin for new cases. Use of MIR: incidence is the measure of risk of disease. For example, after distribution of insecticide-treated nets or indoor spraying in a district, a falling MIR shows that new infections are being prevented, and a rising MIR in a block gives early warning of an outbreak. It is also used in studies of causes of disease.